- Email: [email protected]
Lucky labelings of graphs
Information Processing Letters 109 (2009) 1078–1081
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Lucky labelings of graphs a b ˙ ´ Sebastian Czerwinski , Jarosław Grytczuk b,∗ , Wiktor Zelazny a b
Faculty of Mathematics, Computer Science, and Econometrics, University of Zielona Góra, 65-516 Zielona Góra, Poland Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 8 August 2008 Accepted 12 May 2009 Available online 4 July 2009 Communicated by P.M.B. Vitányi Keywords: Combinatorial problems Lucky labeling Combinatorial Nullstellensatz
a b s t r a c t Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S ( v ) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S (u ) = S ( v ) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1, 2, . . . , k} is the lucky number of G, denoted by η(G ). Using algebraic methods we prove that η(G ) k + 1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η( T ) 2 for every tree T , and η(G ) 3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that η(G ) 100 280 245 065 for every planar graph G. Nevertheless we offer a provocative conjecture that η(G ) χ (G ) for every graph G. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Let f : V (G ) → N be a labeling of the vertices of a graph G by positive integers. Let S ( v ) denote the sum of labels over all neighbors of the vertex v in G. If v is an isolated vertex of G we put S ( v ) = 0. A labeling f is lucky if S (u ) = S ( v ) for every pair of adjacent vertices u and v. The lucky number of a graph G, denoted by η(G ), is the least positive integer k such that G has a lucky labeling with {1, 2, . . . , k} as the set of labels. We are interested in verifying the following conjecture. Conjecture 1. For every graph G the lucky number satisfies
η ( G ) χ ( G ). The idea of lucky labelings arose as a natural vertex ´ version of the problem introduced by Karonski, Łuczak, and Thomason [10]. In this variant a proper coloring of a graph is generated in a similar way via edge labeling,
*
Corresponding author. ´ E-mail addresses: [email protected] (S. Czerwinski), ˙ [email protected] (J. Grytczuk), [email protected] (W. Zelazny). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.05.011
© 2009 Elsevier B.V.
All rights reserved.
where S ( v ) is computed as the sum of labels of all edges incident with v. It is conjectured that three integer labels {1, 2, 3} are sufficient for every connected graph, except K 2 . Currently best result of Addario-Berry et al. [2] guarantees existence of appropriate labeling with sixteen numbers {1, 2, . . . , 16}. See [1,6,11–13] for other variants of the problem, and [8] for a general survey on graph labelings. For vertex labelings the situation is much less clear. First notice that η( K n ) = n, so there is no absolute bound on the lucky number. Actually we are not sure if the lucky number is bounded for graphs of bounded chromatic number, even in the simplest case of bipartite graphs. We give a bound for η(G ) in terms of the acyclic chromatic number of G, which is the least number of colors in a proper vertex coloring of G with additional property that any two color classes span a forest. In particular we get that η(G ) 100 280 245 065 for every planar graph G, which is probably not the best possible bound. For the proof we need a stronger version of the problem, where the labels are taken from a finite abelian group. We show that certain bipartite graphs (including trees and unit interval bigraphs) have lucky labelings modulo n for every n 3. Next, using Combinatorial Nullstellensatz [3] we show that every
S. Czerwi´ nski et al. / Information Processing Letters 109 (2009) 1078–1081
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bipartite graph whose edges can be oriented so that the maximum out-degree of a vertex is at most k has a lucky labeling from lists of size k + 1. We conclude the paper with a short discussion of another algebraic approach.
f (x) = ( f 1 (x), f 2 (x), . . . , f r (x)), where f i (x) = 0 whenever x is not a member of V ( F i ). We claim that f is a lucky labeling with respect to the group operation of Γ (which is coordinatewise addition modulo p i ’s). Indeed, let
2. Induced forests and acyclic chromatic number
S (v ) =
Let n 2 be an integer. We define a lucky labeling modulo n in the same way, except that we compute the sum S ( v ) modulo n. We start with the following simple result for trees.
where N G ( v ) denotes the set of neighbors of v in a graph G. Suppose that v ∈ V ( F i ). Since F i is an induced subgraph of G, every vertex x that is adjacent to v in G but not in F i is not a member of V ( F i ), hence f i (x) = 0. Therefore
Proposition 1. Every tree T has a lucky labeling modulo n for every n 3. Proof. Choose any vertex of T as a root and order the vertices into levels L 0 , L 1 , . . . , L h in a common way, with L 0 consisting of the root only. Color the vertices of T black and white, so that the odd levels are white and the even levels are black, say. Assign the label 0 to every black vertex and assign a list of two non-zero labels to every white vertex. We claim that we can choose labels for white vertices from their lists so that for each black vertex v the sum S ( v ) will be non-zero. This will prove the assertion, as we already have S ( v ) = 0 for every white vertex v. For this purpose we apply induction. Let v 0 , v 1 , . . . , v r be any linear ordering of black vertices so that the vertices from level L i are before the vertices from level L j whenever i < j. Start with assigning labels to the neighbors of the root v 0 from their lists so that the total sum is nonzero modulo n. For the inductive step assume that we have already assigned labels for all neighbors of the vertices v 0 , v 1 , . . . , v j −1 , 0 < j r, so that S ( v i ) = 0 (mod n) for all i = 1, 2, . . . , j − 1. We claim that we can extend the labeling to the whole neighborhood of v j so that S ( v j ) = 0 (mod n). Indeed, if v j is a leaf of T we have nothing to do, as it is adjacent to only one white vertex whose label is non-zero. Otherwise v j has some white neighbors at the next level and these vertices have not been labeled so far. Since each list contains at least two different values, we may chose labels for these vertices so that S ( v j ) = 0 (mod n). This completes the proof. 2 A decomposition of a graph G into induced forests is a coloring of the edges of G such that each color class form an acyclic induced subgraph of a graph G. Theorem 1. Suppose that the edge set of a graph G can be decomposed into r induced forests. Then G has a lucky labeling modulo N = p 1 p 2 . . . p r , where p i are distinct odd prime numbers. Consequently, η(G ) N. Proof. Let F 1 , F 2 , . . . , F r be a collection of induced forests in a graph G such that each edge of G belongs to exactly one of the forests. Assume that p 1 , p 2 , . . . , p r are pairwise distinct odd prime numbers. A forest is just a collection of trees, so, by Proposition 1, each graph F i has a labeling f i : V ( F i ) → Z p i which is a lucky labeling modulo p i . Let Γ = Z p 1 × Z p 2 × · · · × Z pr be the direct product of additive groups Z p i . Now define a labeling f : V (G ) → Γ by
x∈ N G ( v )
f i (x) =
x∈ N F i ( v )
f (x) =
f 1 (x), f 2 (x), . . . , f r (x) ,
x∈ N G ( v )
f i (x).
x∈ N G ( v )
Now any two vertices u and v adjacent in G are adjacent in some F i . Hence their Γ -sums S (u ) and S ( v ) differ on the i-th coordinate, by the assumption that f i was lucky. Notice that Γ is isomorphic to the cyclic group Z N , hence f provides a lucky labeling modulo N. Finally, substituting N for every 0 in labeling f gives a lucky labeling over positive integers with the set of labels from the range {1, 2, . . . , N }. 2 Let a(G ) denote the acyclic chromatic number of a graph G. The above result implies that the lucky number is bounded for graphs of bounded acyclic chromatic number. Corollary 1. Let a(GG)be a graph with acyclic chromatic number a(G ). Let r = 2 and let N = p 1 p 2 . . . p r be the product of r distinct odd primes. Then G has a lucky labeling modulo N. Consequently, η(G ) N. Proof. Let C 1 , C 2 , . . . , C k be the color classes in any acyclic coloring of G. Let F i j be the subgraph of G spanned by the sets C i ∪ C j , with 1 i < j k. Clearly F i j is an induced forest of G and every edge of G belongs to some F i j . 2 Corollary 1 covers many interesting classes of graphs. For instance, by the well-known result of Borodin [7], the acyclic chromatic number of planar graphs is at most 5. Hence
η(G ) 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 = 100 280 245 065 for every planar graph G. Among other classes with bounded lucky number are, for instance, interval graphs of bounded clique number, or more generally, graphs of bounded treewidth. Indeed, it is easy to see that a(G ) k + 1 for any graph G of treewidth at most k. The above approach may be extended in the following way. Let G be the class of bipartite graphs satisfying the statement of Proposition 1, that is, every graph from G has a lucky labeling modulo n for every n 3. Suppose that a graph G has a proper k-coloring such that any two color classes span a subgraph from the class G . Then, arguing similarly as in the proof of Theorem 1, G has a lucky lak beling modulo the product of 2 distinct odd primes. It is perhaps too optimistic to expect that G is actually the class of all bipartite graphs, but maybe at least the following is true.
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Conjecture 2. For every k there is a constant r = r (k) such that every k-colorable graph has a proper r-coloring in which every two color classes span a subgraph from the class G . An interesting class of bipartite graphs contained in G are the unit interval bigraphs (cf. [9]). These graphs are defined as follows. Let V be a finite family of unit-length intervals on the real line. Color the elements of V black and white arbitrarily. Define a graph on the set of vertices V so that uv ∈ E (G ) if and only if u and v intersect and are of different colors. Alternately, every unit interval bigraph can be derived from a unit interval graph by partitioning its vertices arbitrarily into two classes and deleting all edges between vertices from the same class. If H is a subgraph of a graph G we write d H ( v ) for the degree of vertex v in H . We will need the following simple lemma. Lemma 1. Let G be a bipartite graph with color classes A and B. Let H be an induced subgraph of G such that d H ( v ) ∈ {1, 2} for every vertex v ∈ B. Then G has a lucky labeling modulo n for every n 3. Proof. Put f ( v ) = 1 for every vertex v ∈ A ∩ V ( H ) and f ( v ) = 0 for every other vertex of G. Then S ( v ) = 0 for every v ∈ A. Since H is an induced subgraph of G, we have S ( v ) = d H ( v ) = 1 or 2 for every vertex v ∈ B, as desired. 2 Proposition 2. Every unit interval bigraph has a lucky labeling modulo n for every n 3. Proof. Let G be a connected unit interval bigraph with color classes A and B. We just show that G has an induced subgraph H satisfying conditions of Lemma 1. First put all intervals from V (G ) into linear order accordingly to their left ends. Suppose that the first vertex in this order is from the class A (just rename the classes if this is not the case), say a1 . Consider an alternating path P = a1 b1 a2 b2 . . . bn defined so that b i is the furthest neighbor of ai and ai +1 is the farthest neighbor of b i , for i = 1, 2, . . . , n, where bn is the last vertex from the class B. Clearly such a path is uniquely defined. We claim that the subgraph H of G induced by the set B ∪{a1 , a2 , . . . , an } has the desired property. Indeed, since the maximum gap between any two consecutive intervals ai and ai +1 is less than one, every interval b ∈ B is intersected by some ai . Hence deg H (b) 1 for every b ∈ B. Also it is clear that b cannot intersect more than two among ai ’s. Indeed, suppose this is not the case and there is some b intersecting three intervals ai , a j , and ak , for some i < j < k. Since b i intersects ai by definition, it lies to the right of b. But then it must intersect a j and ak also, which contradicts the definition of P . 2 3. Combinatorial Nullstellensatz In this section we present an algebraic approach to the lucky labeling problem. Let G be a graph on the set of vertices V = { v 1 , v 2 , . . . , v n }. Assign variables x1 , x2 , . . . , xn to → − the vertices of G accordingly, and fix some orientation G
of G. Then define the graph polynomial P in these variables by
P (x1 , x2 , . . . , xn ) =
− →
(x j − xi ).
( v i , v j )∈ E ( G )
Notice that choosing different orientations may influence only the sign of the graph polynomial P . Also a graph G is k-colorable if and only if there is substitution for variables in P from any set of k distinct reals giving a non-zero value. We will make use of the following theorem due to Alon [3], known as Combinatorial Nullstellensatz (cf. [4,5], for earlier versions). Theorem 2 (Combinatorial Nullstellensatz). Let F be an arin bitrary field, and let F (x1 , x2 , . . . , xn ) be a polynomial F[x1 , x2 , . . . , xn ]. Suppose the degree deg( F ) of F is ni=1 ki , where each ki is a non-negative integer, and suppose the coeffik k k cient of x11 x22 . . . xnn in F is non-zero. Then, if L 1 , L 2 , . . . , L n are subsets of F with | L i | > ki , there are c 1 ∈ L 1 , c 2 ∈ L 2 , . . . , cn ∈ L n so that F (c 1 , c 2 , . . . , cn ) = 0. For lucky labelings we need a modified polynomial Q (x1 , x2 , . . . , xn ), obtainedfrom the graph polynomial P by substituting the sum v i ∈ N G ( v j ) xi for every variable
x j in P . Clearly, the lucky number of G satisfies η(G ) k if and only if Q (c 1 , c 2 , . . . , cn ) = 0 for some numbers c i ∈ {1, 2, . . . , k}. Theorem 3. Let G be a bipartite graph on n vertices { v 1 , v 2 , . . . , v n }. Let k be the least number such that G has an orientation with maximum out-degree at most k. Let every vertex v of G be assigned with a list L v of at least k + 1 complex numbers. Then there is a lucky labeling f : V (G ) → C of G such that f ( v ) ∈ L v for every v ∈ V (G ). Proof. First orient the edges of G from one color class to − → the other. Denote this orientation by G 1 . Let P (x1 , x2 , . . . , xn ) be the graph polynomial for G with respect to this orientation. Let S v denotes the sum of variables assigned to the neighbors of vertex v. Let Q be the polynomial obtained from P as described above, that is
Q (x1 , x2 , . . . , xn ) =
− →
( S v − S u ).
(u , v )∈ E ( G 1 ) − →
Notice that by the choice of orientation G 1 , each variable appears with the same sign in every term ( S v − S u ) in which it actually appears. Consequently, all monomials of k k k the form x11 x22 . . . xnn , with the same sequence of exponents (k1 , k2 , . . . , kn ), appear with the same sign in the expansion of Q . Thus they cannot cancel. It remains to show that for some monomial in this expansion we will have ki k for all i = 1, 2, . . . , n. To see this fix an ori− → entation G 2 in which maximum out-degree of a vertex is k. Let e = {u , v } be a non-oriented edge of G and let t (e ) − → be the tail of e in the orientation G 2 . Choosing from each term ( S v − S u ) the variable corresponding to t (e ) we get k k k the monomial ±x11 x22 . . . xnn , where ki is the out-degree of − → vertex v i in the orientation G 2 . By Combinatorial Nullstellensatz the assertion of the theorem is proved. 2
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Parameter k from the above theorem is closely related to the edge-density of a graph. Let
ture 1 would follow if we could prove that Q ∈ I k,n implies P ∈ I k,n .
L (G ) = max
Acknowledgement
| E ( H )| : H is a subgraph of G . | V ( H )|
Then G has an orientation with maximum out-degree at most k if and only if L (G ) k (cf. [5]). In particular, we get that η( T ) 2 for every tree T . We also have the following corollary. Corollary 2. Every bipartite planar graph G has a lucky labeling from lists of size 3. In particular, η(G ) 3. 4. Final comments There are several heuristic arguments supporting Conjecture 1, or at least a weaker supposition that the lucky number is bounded for graphs of bounded chromatic number. For instance it is easy to see that every graph G has a lucky labeling from any set of at most χ (G ) reals provided the set is independent over the rationals. Also, every graph has a lucky labeling from some set of χ (G ) positive integers with sufficiently large gaps (depending on the maximum degree of G). Let I k,n be the ideal in the ring C[x1 , x2 , . . . , xn ] generated by n polynomials of the form (xi − 1)(xi − 2) . . . (xi − k), i = 1, 2, . . . , n. Similarly as in [3] (Theorem 9.3) one may prove that a graph G of order n satisfies η(G ) > k if and only if its polynomial Q (x1 , x2 , . . . , xn ) ∈ I k,n . The same argument gives that χ (G ) > k if and only if the graph polynomial P (x1 , x2 , . . . , xn ) belongs to I k,n . Hence, Conjec-
´ Sebastian Czerwinski acknowledges a support from EU EFS project: “Stypendia dla studentów studiów doktoranckich na WMIiE UZ”. References [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal, B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory Ser. B 94 (2005) 237–244. [2] L. Addario-Berry, K. Dalal, B.A. Reed, Degree constrained subgraphs, Discrete Appl. Math. 156 (7) (2008) 1168–1174. [3] N. Alon, Combinatorial Nullstellensatz, Combin. Prob. Comput. 8 (1999) 7–29. [4] N. Alon, S. Friedland, G. Kalai, Regular subgraphs of almost regular graphs, J. Combin. Theory Ser. B 37 (1984) 79–91. [5] N. Alon, M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125–134. [6] T. Bartnicki, J. Grytczuk, S. Niwczyk, Weight choosability of graphs, J. Graph Theory 60 (2009) 242–256. [7] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211–236. [8] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. DS 6 (2008), 219 pp. [9] P. Hell, J. Huang, Interval bigraphs and circular arc graphs, J. Graph Theory 46 (2004) 313–327. ´ [10] M. Karonski, T. Łuczak, A. Thomason, Edge weights and vertex colours, J. Combin. Theory Ser. B 91 (2004) 151–157. [11] J. Przybyło, M. Wozniak, ´ 1, 2 conjecture, Preprint MD 024, available at http://www.ii.uj.edu.pl/preMD/index.php. [12] J. Przybyło, M. Wozniak, ´ 1, 2 conjecture, Preprint MD 026, available at http://www.ii.uj.edu.pl/preMD/index.php. [13] J. Sowronek-Kaziów, 1, 2 conjecture – the multiplicative version, Inform. Process. Lett. 107 (2008) 93–95.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Lucky labelings of graphs a b ˙ ´ Sebastian Czerwinski , Jarosław Grytczuk b,∗ , Wiktor Zelazny a b
Faculty of Mathematics, Computer Science, and Econometrics, University of Zielona Góra, 65-516 Zielona Góra, Poland Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 8 August 2008 Accepted 12 May 2009 Available online 4 July 2009 Communicated by P.M.B. Vitányi Keywords: Combinatorial problems Lucky labeling Combinatorial Nullstellensatz
a b s t r a c t Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S ( v ) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S (u ) = S ( v ) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1, 2, . . . , k} is the lucky number of G, denoted by η(G ). Using algebraic methods we prove that η(G ) k + 1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η( T ) 2 for every tree T , and η(G ) 3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that η(G ) 100 280 245 065 for every planar graph G. Nevertheless we offer a provocative conjecture that η(G ) χ (G ) for every graph G. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Let f : V (G ) → N be a labeling of the vertices of a graph G by positive integers. Let S ( v ) denote the sum of labels over all neighbors of the vertex v in G. If v is an isolated vertex of G we put S ( v ) = 0. A labeling f is lucky if S (u ) = S ( v ) for every pair of adjacent vertices u and v. The lucky number of a graph G, denoted by η(G ), is the least positive integer k such that G has a lucky labeling with {1, 2, . . . , k} as the set of labels. We are interested in verifying the following conjecture. Conjecture 1. For every graph G the lucky number satisfies
η ( G ) χ ( G ). The idea of lucky labelings arose as a natural vertex ´ version of the problem introduced by Karonski, Łuczak, and Thomason [10]. In this variant a proper coloring of a graph is generated in a similar way via edge labeling,
*
Corresponding author. ´ E-mail addresses: [email protected] (S. Czerwinski), ˙ [email protected] (J. Grytczuk), [email protected] (W. Zelazny). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.05.011
© 2009 Elsevier B.V.
All rights reserved.
where S ( v ) is computed as the sum of labels of all edges incident with v. It is conjectured that three integer labels {1, 2, 3} are sufficient for every connected graph, except K 2 . Currently best result of Addario-Berry et al. [2] guarantees existence of appropriate labeling with sixteen numbers {1, 2, . . . , 16}. See [1,6,11–13] for other variants of the problem, and [8] for a general survey on graph labelings. For vertex labelings the situation is much less clear. First notice that η( K n ) = n, so there is no absolute bound on the lucky number. Actually we are not sure if the lucky number is bounded for graphs of bounded chromatic number, even in the simplest case of bipartite graphs. We give a bound for η(G ) in terms of the acyclic chromatic number of G, which is the least number of colors in a proper vertex coloring of G with additional property that any two color classes span a forest. In particular we get that η(G ) 100 280 245 065 for every planar graph G, which is probably not the best possible bound. For the proof we need a stronger version of the problem, where the labels are taken from a finite abelian group. We show that certain bipartite graphs (including trees and unit interval bigraphs) have lucky labelings modulo n for every n 3. Next, using Combinatorial Nullstellensatz [3] we show that every
S. Czerwi´ nski et al. / Information Processing Letters 109 (2009) 1078–1081
1079
bipartite graph whose edges can be oriented so that the maximum out-degree of a vertex is at most k has a lucky labeling from lists of size k + 1. We conclude the paper with a short discussion of another algebraic approach.
f (x) = ( f 1 (x), f 2 (x), . . . , f r (x)), where f i (x) = 0 whenever x is not a member of V ( F i ). We claim that f is a lucky labeling with respect to the group operation of Γ (which is coordinatewise addition modulo p i ’s). Indeed, let
2. Induced forests and acyclic chromatic number
S (v ) =
Let n 2 be an integer. We define a lucky labeling modulo n in the same way, except that we compute the sum S ( v ) modulo n. We start with the following simple result for trees.
where N G ( v ) denotes the set of neighbors of v in a graph G. Suppose that v ∈ V ( F i ). Since F i is an induced subgraph of G, every vertex x that is adjacent to v in G but not in F i is not a member of V ( F i ), hence f i (x) = 0. Therefore
Proposition 1. Every tree T has a lucky labeling modulo n for every n 3. Proof. Choose any vertex of T as a root and order the vertices into levels L 0 , L 1 , . . . , L h in a common way, with L 0 consisting of the root only. Color the vertices of T black and white, so that the odd levels are white and the even levels are black, say. Assign the label 0 to every black vertex and assign a list of two non-zero labels to every white vertex. We claim that we can choose labels for white vertices from their lists so that for each black vertex v the sum S ( v ) will be non-zero. This will prove the assertion, as we already have S ( v ) = 0 for every white vertex v. For this purpose we apply induction. Let v 0 , v 1 , . . . , v r be any linear ordering of black vertices so that the vertices from level L i are before the vertices from level L j whenever i < j. Start with assigning labels to the neighbors of the root v 0 from their lists so that the total sum is nonzero modulo n. For the inductive step assume that we have already assigned labels for all neighbors of the vertices v 0 , v 1 , . . . , v j −1 , 0 < j r, so that S ( v i ) = 0 (mod n) for all i = 1, 2, . . . , j − 1. We claim that we can extend the labeling to the whole neighborhood of v j so that S ( v j ) = 0 (mod n). Indeed, if v j is a leaf of T we have nothing to do, as it is adjacent to only one white vertex whose label is non-zero. Otherwise v j has some white neighbors at the next level and these vertices have not been labeled so far. Since each list contains at least two different values, we may chose labels for these vertices so that S ( v j ) = 0 (mod n). This completes the proof. 2 A decomposition of a graph G into induced forests is a coloring of the edges of G such that each color class form an acyclic induced subgraph of a graph G. Theorem 1. Suppose that the edge set of a graph G can be decomposed into r induced forests. Then G has a lucky labeling modulo N = p 1 p 2 . . . p r , where p i are distinct odd prime numbers. Consequently, η(G ) N. Proof. Let F 1 , F 2 , . . . , F r be a collection of induced forests in a graph G such that each edge of G belongs to exactly one of the forests. Assume that p 1 , p 2 , . . . , p r are pairwise distinct odd prime numbers. A forest is just a collection of trees, so, by Proposition 1, each graph F i has a labeling f i : V ( F i ) → Z p i which is a lucky labeling modulo p i . Let Γ = Z p 1 × Z p 2 × · · · × Z pr be the direct product of additive groups Z p i . Now define a labeling f : V (G ) → Γ by
x∈ N G ( v )
f i (x) =
x∈ N F i ( v )
f (x) =
f 1 (x), f 2 (x), . . . , f r (x) ,
x∈ N G ( v )
f i (x).
x∈ N G ( v )
Now any two vertices u and v adjacent in G are adjacent in some F i . Hence their Γ -sums S (u ) and S ( v ) differ on the i-th coordinate, by the assumption that f i was lucky. Notice that Γ is isomorphic to the cyclic group Z N , hence f provides a lucky labeling modulo N. Finally, substituting N for every 0 in labeling f gives a lucky labeling over positive integers with the set of labels from the range {1, 2, . . . , N }. 2 Let a(G ) denote the acyclic chromatic number of a graph G. The above result implies that the lucky number is bounded for graphs of bounded acyclic chromatic number. Corollary 1. Let a(GG)be a graph with acyclic chromatic number a(G ). Let r = 2 and let N = p 1 p 2 . . . p r be the product of r distinct odd primes. Then G has a lucky labeling modulo N. Consequently, η(G ) N. Proof. Let C 1 , C 2 , . . . , C k be the color classes in any acyclic coloring of G. Let F i j be the subgraph of G spanned by the sets C i ∪ C j , with 1 i < j k. Clearly F i j is an induced forest of G and every edge of G belongs to some F i j . 2 Corollary 1 covers many interesting classes of graphs. For instance, by the well-known result of Borodin [7], the acyclic chromatic number of planar graphs is at most 5. Hence
η(G ) 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 = 100 280 245 065 for every planar graph G. Among other classes with bounded lucky number are, for instance, interval graphs of bounded clique number, or more generally, graphs of bounded treewidth. Indeed, it is easy to see that a(G ) k + 1 for any graph G of treewidth at most k. The above approach may be extended in the following way. Let G be the class of bipartite graphs satisfying the statement of Proposition 1, that is, every graph from G has a lucky labeling modulo n for every n 3. Suppose that a graph G has a proper k-coloring such that any two color classes span a subgraph from the class G . Then, arguing similarly as in the proof of Theorem 1, G has a lucky lak beling modulo the product of 2 distinct odd primes. It is perhaps too optimistic to expect that G is actually the class of all bipartite graphs, but maybe at least the following is true.
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Conjecture 2. For every k there is a constant r = r (k) such that every k-colorable graph has a proper r-coloring in which every two color classes span a subgraph from the class G . An interesting class of bipartite graphs contained in G are the unit interval bigraphs (cf. [9]). These graphs are defined as follows. Let V be a finite family of unit-length intervals on the real line. Color the elements of V black and white arbitrarily. Define a graph on the set of vertices V so that uv ∈ E (G ) if and only if u and v intersect and are of different colors. Alternately, every unit interval bigraph can be derived from a unit interval graph by partitioning its vertices arbitrarily into two classes and deleting all edges between vertices from the same class. If H is a subgraph of a graph G we write d H ( v ) for the degree of vertex v in H . We will need the following simple lemma. Lemma 1. Let G be a bipartite graph with color classes A and B. Let H be an induced subgraph of G such that d H ( v ) ∈ {1, 2} for every vertex v ∈ B. Then G has a lucky labeling modulo n for every n 3. Proof. Put f ( v ) = 1 for every vertex v ∈ A ∩ V ( H ) and f ( v ) = 0 for every other vertex of G. Then S ( v ) = 0 for every v ∈ A. Since H is an induced subgraph of G, we have S ( v ) = d H ( v ) = 1 or 2 for every vertex v ∈ B, as desired. 2 Proposition 2. Every unit interval bigraph has a lucky labeling modulo n for every n 3. Proof. Let G be a connected unit interval bigraph with color classes A and B. We just show that G has an induced subgraph H satisfying conditions of Lemma 1. First put all intervals from V (G ) into linear order accordingly to their left ends. Suppose that the first vertex in this order is from the class A (just rename the classes if this is not the case), say a1 . Consider an alternating path P = a1 b1 a2 b2 . . . bn defined so that b i is the furthest neighbor of ai and ai +1 is the farthest neighbor of b i , for i = 1, 2, . . . , n, where bn is the last vertex from the class B. Clearly such a path is uniquely defined. We claim that the subgraph H of G induced by the set B ∪{a1 , a2 , . . . , an } has the desired property. Indeed, since the maximum gap between any two consecutive intervals ai and ai +1 is less than one, every interval b ∈ B is intersected by some ai . Hence deg H (b) 1 for every b ∈ B. Also it is clear that b cannot intersect more than two among ai ’s. Indeed, suppose this is not the case and there is some b intersecting three intervals ai , a j , and ak , for some i < j < k. Since b i intersects ai by definition, it lies to the right of b. But then it must intersect a j and ak also, which contradicts the definition of P . 2 3. Combinatorial Nullstellensatz In this section we present an algebraic approach to the lucky labeling problem. Let G be a graph on the set of vertices V = { v 1 , v 2 , . . . , v n }. Assign variables x1 , x2 , . . . , xn to → − the vertices of G accordingly, and fix some orientation G
of G. Then define the graph polynomial P in these variables by
P (x1 , x2 , . . . , xn ) =
− →
(x j − xi ).
( v i , v j )∈ E ( G )
Notice that choosing different orientations may influence only the sign of the graph polynomial P . Also a graph G is k-colorable if and only if there is substitution for variables in P from any set of k distinct reals giving a non-zero value. We will make use of the following theorem due to Alon [3], known as Combinatorial Nullstellensatz (cf. [4,5], for earlier versions). Theorem 2 (Combinatorial Nullstellensatz). Let F be an arin bitrary field, and let F (x1 , x2 , . . . , xn ) be a polynomial F[x1 , x2 , . . . , xn ]. Suppose the degree deg( F ) of F is ni=1 ki , where each ki is a non-negative integer, and suppose the coeffik k k cient of x11 x22 . . . xnn in F is non-zero. Then, if L 1 , L 2 , . . . , L n are subsets of F with | L i | > ki , there are c 1 ∈ L 1 , c 2 ∈ L 2 , . . . , cn ∈ L n so that F (c 1 , c 2 , . . . , cn ) = 0. For lucky labelings we need a modified polynomial Q (x1 , x2 , . . . , xn ), obtainedfrom the graph polynomial P by substituting the sum v i ∈ N G ( v j ) xi for every variable
x j in P . Clearly, the lucky number of G satisfies η(G ) k if and only if Q (c 1 , c 2 , . . . , cn ) = 0 for some numbers c i ∈ {1, 2, . . . , k}. Theorem 3. Let G be a bipartite graph on n vertices { v 1 , v 2 , . . . , v n }. Let k be the least number such that G has an orientation with maximum out-degree at most k. Let every vertex v of G be assigned with a list L v of at least k + 1 complex numbers. Then there is a lucky labeling f : V (G ) → C of G such that f ( v ) ∈ L v for every v ∈ V (G ). Proof. First orient the edges of G from one color class to − → the other. Denote this orientation by G 1 . Let P (x1 , x2 , . . . , xn ) be the graph polynomial for G with respect to this orientation. Let S v denotes the sum of variables assigned to the neighbors of vertex v. Let Q be the polynomial obtained from P as described above, that is
Q (x1 , x2 , . . . , xn ) =
− →
( S v − S u ).
(u , v )∈ E ( G 1 ) − →
Notice that by the choice of orientation G 1 , each variable appears with the same sign in every term ( S v − S u ) in which it actually appears. Consequently, all monomials of k k k the form x11 x22 . . . xnn , with the same sequence of exponents (k1 , k2 , . . . , kn ), appear with the same sign in the expansion of Q . Thus they cannot cancel. It remains to show that for some monomial in this expansion we will have ki k for all i = 1, 2, . . . , n. To see this fix an ori− → entation G 2 in which maximum out-degree of a vertex is k. Let e = {u , v } be a non-oriented edge of G and let t (e ) − → be the tail of e in the orientation G 2 . Choosing from each term ( S v − S u ) the variable corresponding to t (e ) we get k k k the monomial ±x11 x22 . . . xnn , where ki is the out-degree of − → vertex v i in the orientation G 2 . By Combinatorial Nullstellensatz the assertion of the theorem is proved. 2
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Parameter k from the above theorem is closely related to the edge-density of a graph. Let
ture 1 would follow if we could prove that Q ∈ I k,n implies P ∈ I k,n .
L (G ) = max
Acknowledgement
| E ( H )| : H is a subgraph of G . | V ( H )|
Then G has an orientation with maximum out-degree at most k if and only if L (G ) k (cf. [5]). In particular, we get that η( T ) 2 for every tree T . We also have the following corollary. Corollary 2. Every bipartite planar graph G has a lucky labeling from lists of size 3. In particular, η(G ) 3. 4. Final comments There are several heuristic arguments supporting Conjecture 1, or at least a weaker supposition that the lucky number is bounded for graphs of bounded chromatic number. For instance it is easy to see that every graph G has a lucky labeling from any set of at most χ (G ) reals provided the set is independent over the rationals. Also, every graph has a lucky labeling from some set of χ (G ) positive integers with sufficiently large gaps (depending on the maximum degree of G). Let I k,n be the ideal in the ring C[x1 , x2 , . . . , xn ] generated by n polynomials of the form (xi − 1)(xi − 2) . . . (xi − k), i = 1, 2, . . . , n. Similarly as in [3] (Theorem 9.3) one may prove that a graph G of order n satisfies η(G ) > k if and only if its polynomial Q (x1 , x2 , . . . , xn ) ∈ I k,n . The same argument gives that χ (G ) > k if and only if the graph polynomial P (x1 , x2 , . . . , xn ) belongs to I k,n . Hence, Conjec-
´ Sebastian Czerwinski acknowledges a support from EU EFS project: “Stypendia dla studentów studiów doktoranckich na WMIiE UZ”. References [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal, B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory Ser. B 94 (2005) 237–244. [2] L. Addario-Berry, K. Dalal, B.A. Reed, Degree constrained subgraphs, Discrete Appl. Math. 156 (7) (2008) 1168–1174. [3] N. Alon, Combinatorial Nullstellensatz, Combin. Prob. Comput. 8 (1999) 7–29. [4] N. Alon, S. Friedland, G. Kalai, Regular subgraphs of almost regular graphs, J. Combin. Theory Ser. B 37 (1984) 79–91. [5] N. Alon, M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125–134. [6] T. Bartnicki, J. Grytczuk, S. Niwczyk, Weight choosability of graphs, J. Graph Theory 60 (2009) 242–256. [7] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211–236. [8] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. DS 6 (2008), 219 pp. [9] P. Hell, J. Huang, Interval bigraphs and circular arc graphs, J. Graph Theory 46 (2004) 313–327. ´ [10] M. Karonski, T. Łuczak, A. Thomason, Edge weights and vertex colours, J. Combin. Theory Ser. B 91 (2004) 151–157. [11] J. Przybyło, M. Wozniak, ´ 1, 2 conjecture, Preprint MD 024, available at http://www.ii.uj.edu.pl/preMD/index.php. [12] J. Przybyło, M. Wozniak, ´ 1, 2 conjecture, Preprint MD 026, available at http://www.ii.uj.edu.pl/preMD/index.php. [13] J. Sowronek-Kaziów, 1, 2 conjecture – the multiplicative version, Inform. Process. Lett. 107 (2008) 93–95.